 Bernstein and Maximal InequalitiesJunwei Lu
Chapter Chapter 6 in Big Data Analysis, 2025, pp 29-38 from Springer Abstract: Abstract In the previous chapter, we showed that the sample mean of independent sub-exponential random variables X 1 , … , X n $$X_1, \ldots, X_n$$ with the parameter α $$\alpha $$ has the tail probability ℙ ( | X ̄ n − 𝔼 X | > t ) ≤ = 2 e − n 2 t 2 α 2 ∧ t α , $$\displaystyle \mathbb {P} (|\bar {X}_n - \mathbb {E} X| > t) \le = 2e^{- \frac {n}{2} \left (\frac {t^2}{\alpha ^2} \wedge \frac {t}{\alpha} \right)}, $$ where x ∧ y = min ( x , y ) $$x \wedge y = \min (x, y)$$ and x ∨ y = max ( x , y ) $$x \vee y = \max (x, y)$$ . Therefore, with probability at least 1 − δ $$1- \delta $$ , | X ̄ n − 𝔼 X | ≤ α 2 n log ( 2 δ ) ∨ α n log ( 2 δ ) . $$\displaystyle \lvert \bar {X}_n - \mathbb {E} X \rvert \le \sqrt {\frac {\alpha ^2}{n} \log \Big (\frac {2}{\delta}}\Big) \vee \left (\frac {\alpha}{n} \log \Big (\frac {2}{\delta}\Big) \right). $$ We can see that the two types of sup-exponential tail probability give us two types of rate: O ( α ∕ n ) $$O(\alpha / \sqrt {n})$$ and O ( α ∕ n ) $$O(\alpha / n)$$ . Although the second term is dominated by the first term, it implies the possibility of giving two types of rates in the concentration inequality. We are going to show a stronger concentration inequality of such type. Date: 2025 There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it. Related works: Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-032-03161-7_6 Ordering information: This item can be ordered from DOI: 10.1007/978-3-032-03161-7_6 Access Statistics for this chapter More chapters in Springer Books from Springer |
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